Spacing properties of the zeros of orthogonal polynomials on Cantor sets via a sequence of polynomial mappings

TL;DR

This paper investigates the spacing of zeros of orthogonal polynomials on Cantor sets, establishing bounds based on polynomial mappings and critical points, with implications for measures with singular continuous spectra.

Contribution

It introduces a novel approach linking polynomial mappings and critical points to zero spacing, providing sharp bounds for orthogonal polynomials on Cantor sets.

Findings

Lower bounds on zero spacing based on critical points

Sharp bounds for zeros of orthogonal polynomials on singular measures

Connection between polynomial mappings and zero distribution

Abstract

Let $\mu$ be a probability measure with an infinite compact support on $\mathbb{R}$. Let us further assume that $(F_n)_{n=1}^\infty$ is a sequence of orthogonal polynomials for $\mu$ where $(f_n)_{n=1}^\infty$ is a sequence of nonlinear polynomials and $F_n:=f_n\circ\dots\circ f_1$ for all $n\in\mathbb{N}$. We prove that if there is an $s_0\in\mathbb{N}$ such that $0$ is a root of $f_n^\prime$ for each $n>s_0$ then the distance between any two zeros of an orthogonal polynomial for $\mu$ of a given degree greater than $1$ has a lower bound in terms of the distance between the set of critical points and the set of zeros of some $F_k$. Using this, we find sharp bounds from below and above for the infimum of distances between the consecutive zeros of orthogonal polynomials for singular continuous measures.